Splung.com Physics Forum

[kwantum]

A wall clock on Planet X has two hands that are aligned at midnight and turn in the same direction at uniform rates, one at 0.0425 rad/s and the other at 0.0163 rad/s. At how many seconds after midnight are these hands (a) first aligned and (b) next aligned?
I am having trouble figuring out how to set up the problem to find the answers.

[hepcj]

The clock hands travel at different rates. Call them w1 = 0.0163 rad/s and w2 = 0.0425 rad/s.
As both hands are lined up when they start and the one hand travels faster than the other then there is no opportunity for them to align on the first time around.
By the time the faster hand comes around to the start it will have travelled 2 Pi rads. In that time the second hand will also have moved on by an amount 2pi*(w1/w2). By the time the first hand moves through this angle the second hand will have moved on by (w1/w2)*(w1/w2)*2 pi. By the time the first hand gets to this point the second hand will have moved and so on.
We can represent the point where the hands align by an infinite sum of powers of (w1/w2).
ie
Sum[(w1/w2)^i] where i = 0,1,2.... infinity

luckily this reduces to (w2/(w2-w1))

multiply this by the time it takes to go around once (2pi/w2)

and this gives 239.8 seconds.

Using the circular symmetry of the clock, we can say that it will also take as long before they meet again so the second alignment will occur at 479.6 seconds past midnight.